> On 12/04/2013 7:50 AM, Alan Smaill wrote:>> Nam Nguyen <namducnguyen@shaw.ca> writes:>>>>> On 12/04/2013 3:29 AM, Alan Smaill wrote:>>>> Nam Nguyen <namducnguyen@shaw.ca> writes:>>>>>>>>> But what is "meta-logic of the meta-language", in the context of FOL>>>>> structure? Or is that at best just intuition and at worst just a>>>>> buzzword?>>>>>>>> You tell us that it is possible to reason about language structures.>>>> What logic are you using to do that -- or is that at best just intuition?>>>>>> I've used FOL ( _First Order Logic_ ) definitions that one should be>>> familiar with.>>>>>> If anything, notation like "this" is defined in term of FOL>>> terminologies.>>>> ie, you use a first-order meta-logic.>> OK. Why don't we just call it First Order Logic as everyone is supposed> to understand already?

Because you now have two logics --good old statements like cGC are not statements about syntax,they are statements about numbers. The language structures foryour theory of syntax is different from language structures fornatural numbers.

>>> So I don't see all that historical context of "meta-logic" would>>> have anything to do with the issue of, say, whether or not it's>>> impossible to construct the naturals as a language model.>>>> The strength of the meta-logic does have a bearing.>> Do you allow proof by induction over the syntax of formulas,>> for example?>> So you're talking the strength (and weakness) of First Order Logic.

Not only -- see above.But do you allow proof by induction over the syntax of formulas?

>>> If you could construct it, as I did construct Mg, M1, ..., then present>>> the construction, otherwise at least for the time being admit you>>> couldn't do it.>>>>>> Why would that be such a difficult task for one to do?>>>> That's not what's at issue here.>>>> Suppose you have a language structure for the language of Peano>> Arithmetic where all the axioms of PA are true, and suppose>> that the underlying set is X, and "0" is the constant used>> for the number zero.>> "Suppose" is hypothetical but sure, I could hypothesize such.

How could it not be the case?

>> Is there a way of characterising the element of Z correponding to "0"?>> Or is that a matter of opinion?>>> But, FOL definition of language structure already provides that, e.g.:>> ('0', {}).>> Right?

If you're happy with that, fine.

Now can you characterise the element of X that corresponds to "S(0)",with "S" the symbol for successor?